Integral Hodge classes on fourfolds fibered by quadric bundles

Abstract

We discuss the space of sections and certain bisections on a quadric surfaces bundle X X over a smooth curve. The Abel-Jacobi from these spaces to the intermediate Jacobian will be shown to be dominant with rationally connected fibers. As an application, we prove that the integral Hodge conjecture holds for degree 4 4 integral Hodge classes (IHC) of fourfolds fibered by quadric bundles over a smooth curve. This gives an alternative proof of a result of Colliot-Thélène and Voisin.